Preheat your oven to 325°F and line two baking sheets with parchment paper. In a large bowl, beat the softened butter with powdered sugar until light and creamy. Stir in the lemon zest.
Position the dough between two lightly floured sheets of parchment paper. Roll it to a thickness of approximately 1/4 inch and use your preferred cookie cutters to cut out shapes.
Arrange the cut cookies on the prepared baking sheets and bake for 8 to 11 minutes, or until the edges are a light golden brown
In a small bowl, whisk together sifted powdered sugar and lemon juice to form the glaze. Add more lemon juice if needed to reach desired thickness.